Generalized Hukuhara-Clarke Derivative of Interval-valued Functions and its Properties

In this article, the notion of gH-Clarke derivative for interval-valued functions is proposed. To define the concept of gH-Clarke derivatives, the concepts of limit superior, limit inferior, and sublinear interval-valued functions are studied in the sequel. The upper gH-Clarke derivative of a gH-Lipschitz interval-valued function (IVF) is observed to be a sublinear IVF. It is found that every gH-Lipschitz continuous function is upper gH-Clarke differentiable. For a convex and gH-Lipschitz IVF, it is shown that the upper gH-Clarke derivative coincides with the gH-directional derivative. The entire study is supported by suitable illustrative examples.


Introduction
Clarke derivative [10] is applied in the nonsmooth analysis where the functions do not have a unique linear approximation. Advances of nonsmooth analysis [7,33] show the essential need of this derivative to handle nondifferentiable functions, especially in the absence of convexity. The topics of optimization [21], control theory [21], variational method [1], etc. are wide application areas of Clarke derivative.
As the topic of this study is Clarke derivatives for IVFs, in the following subsection, we describe the literature on Interval Optimization Problems (IOPs) and calculus of Interval-Valued Functions (IVFs). The analysis of IVFs enables one to effectively deal with the errors/uncertainties that appear while modeling practical problems. It is to be noted that there is a relatively large joint intersection of the literature survey of this paper with the recent paper by Ghosh et al. [16].
In the study of interval analysis, in addition to interval arithmetic, an appropriate ordering of intervals and the calculus of IVFs play key roles. Unlike the real numbers, intervals are not linearly ordered. Thus, the development of optimization theory with interval-valued function is not a trivial extension of the conventional optimization theory. Most often [6,16,36,40], IOPs have been analyzed with respect to a partial ordering [20]. Some researchers [3,12] used ordering relations of intervals based on the parametric comparison of intervals. In [8], an ordering relation of intervals is defined by a bijective map from the set of intervals to the Euclidean plane R 2 . However, these ordering relations [3,12,8] of intervals can be derived from the relations described in [20]. Sengupta et al. [34] proposed an acceptability function for intervals, just like a fuzzy membership function. Recently, Ghosh et al. [17] investigated variable ordering relations for intervals and used them in IOPs.
To observe the properties of an IVF, calculus plays an essential role. Initially, in order to develop the calculus of IVFs, Hukuhara [19] introduced the concept of differentiability of IVFs with the help of Hukuhara difference of intervals. However, the definition of Hukuhara differentiability is restrictive [6]. In general, if F(x) = A ⊙ f (x), where A is a compact interval and f (x) is a real-valued function, then F is not Hukuhara differentiable in the case of f ′ (x) < 0 [2]. In order to refine the calculus of IVFs, the concepts of gH-derivative, gH-partial derivative, gH-gradient, and gH-differentiability for IVFs have been developed in [6,11,27,36,37]. Recently, Ghosh et al. [16] have provided the idea of gH-directional derivative, gH-Gâteaux derivative, and gH-Fréchet derivative of IVFs.

Motivation and Contribution
From the literature on the analysis of IVFs, one can notice that the study of traditional generalized derivative (Clarke derivative) for IVFs have not been developed so far. However, the basic properties of generalized derivatives might be beneficial for characterizing and capturing the optimal solutions of IOPs with nonsmooth IVFs. To define and find properties of Clarke derivative of IVFs, we need to establish the notions of limit superior and sublinearity for IVFs. In this article, after illustrating the concept of limit superior, limit inferior, and sublinearity of IVFs, we define upper and lower gH-Clarke derivatives of IVFs. Although both of the upper and lower gH-Clarke derivatives of IVFs are defined in this article, only the properties of the upper gH-Clarke derivative are studied since the results for the lower derivative can be used analogously. It is shown that if an IVF is upper gH-Clarke differentiable at a point, then its derivative is a sublinear IVF. We further prove that the upper gH-Clarke derivative exists at any point if the IVF is convex gH-Lipschitz and the derivative is equal to gH-directional derivative.

Delineation
The rest of the article is demonstrated in the following sequence. The next section covers some basic terminology and notions of convex analysis and interval analysis, followed by the convexity and calculus of IVFs that are required in this paper. Also, a few properties of intervals, the gH-directional of an IVF is discussed in Section 2. The concepts of limit superior, sublinear IVF and their properties are given in Section 3. In the same section, we define upper gH-Clarke derivative, lower gH-Clarke derivative of IVFs, and prove that the upper gH-Clarke derivative of gH-Lipschitz IVF always exists. Also, for convex gH-Lipschitz continuous IVF, it is shown that upper gH-Clarke derivative coincides with gH-directional derivative. Further, the sublinearity of the upper gH-Clarke derivative is shown in the same section.

Preliminaries and Terminology
This section is devoted to some basic notions on intervals and the convexity and calculus of IVFs. Throughout the paper, we use the following notations.
• X denotes a real normed linear space with the norm · • B(x, δ) denotes the open ball centered atx ∈ X with radius δ • R denotes the set of real numbers • R + denotes the set of nonnegative real numbers

Arithmetic of Intervals and their Dominance Relation
In this section, we discuss Moore's interval arithmetic [28,29] followed by the concepts of gH-difference of two intervals and ordering of intervals [20].
Throughout the article, we denote the set of closed and bounded intervals by I(R) and the elements of I(R) by bold capital letters: A, B, C, . . .. We represent an element A of I(R) in its interval form with the help of the corresponding small letter in the following way: , where a and a are real numbers such that a ≤ a.
It is noteworthy that any singleton {p} of R or, a real number can be represented by an interval P = [p, p], where p = p = p. In particular,

Consider two intervals
The subtraction of B from A, denoted by A ⊖ B, is defined by The multiplication of A and B, denoted by A ⊙ B, is defined by A ⊙ B = min a b, ab, ab, ab , max a b, ab, ab, ab .
The multiplication of A by a real constant λ, denoted by λ ⊙ A or A ⊙ λ, is defined by Notice that the definition of λ ⊙ A follows from the fact λ = [λ, λ] and the definition of multiplication A ⊙ B.
Let 0 ∈ B. The division of A by B, denoted by A ⊘ B, is defined by Since A ⊖ A = 0 for any nondegenerate interval A, we use the following concept of difference of intervals in this article.
Definition 2.1. (gH-difference of intervals [35]). Let A = [a, a] and B = [b, b] be two elements of I(R). The gH-difference between A and B, denoted by A ⊖ gH B, is defined by the interval C such that It is to be noted that for A = [a, a] and B = b, b , In the following, we provide a domination relation on intervals that is used throughout the paper. We remark that domination in the following definition is based on a minimization type optimization problems: the smaller value the better.  (ii) B is said to be strictly dominated by A if either 'a ≤ b and a < b' or 'a < b and a ≤ b', and then we write A ≺ B;

Few Properties of Intervals
In this subsection, a few properties of the elements of I(R) is studied that are used later in the paper. In the rest of the paper, by the norm of an interval, we refer to the following definition.  For all x, y ∈ R and C ∈ I(R), Proof. See Appendix A.
Lemma 2.2. For all A, B, C, D ∈ I(R), Proof. See Appendix B.
Remark 1. The following two points are noticeable.
Therefore, (ii) of Lemma 2.2 is not a trivial property.
Hence, B ⊖ gH A C, but B and A ⊕ C are not comparable. Thus, (ii) of Lemma 2.2 is not an obvious property.

Convexity and Calculus of IVFs
A function F from a nonempty subset S of X to I(R) is known as an IVF. For each argument point x ∈ S, F can be presented by intervals The IVF F is said to be gH-continuous [11] at an interior pointx ∈ S if If F is gH-continuous at each x in S, then F is said to be gH-continuous on S. The IVF F is said to be gH-Lipschitz continuous [16] atx ∈ S if there exist constants K ′ > 0 and δ > 0 such that The constant K ′ is called a Lipschitz constant of F atx. If there exists a K > 0 such that then the IVF F is said to be gH-Lipschitz continuous on S and the constant K is said to be a Lipschitz constant of F on S.  (iii) If F is a gH-Lipschitz continuous on S, then F is gH-continuous on S.

Proof. See Appendix C.
A consequence of Lemma 2.4 is that gH-continuity and gH-Lipschitz continuity of IVFs can be defined classically, i.e., without the concept of gH-difference. Then, the prefix gH-in continuity and Lipschitz continuity could be omitted.

Remark 2.
Converse of (iii) of Lemma 2.4 is not true. For example, consider X as the Euclidean space R, S = [0, 10], and the IVF F : S → I(R), which is defined by  [16,37]). Let F be an IVF on a nonempty subset S of X . Letx ∈ S and h ∈ X . If the limit exists finitely, then the limit is said to be gH-directional derivative of F atx in the direction h, and it is denoted by F D (x)(h).

gH-Clarke Derivative of IVF
In this section, extended concepts of the gH-directional derivative, namely upper and lower gH-Clarke derivatives, for IVFs are given. A short discussion of the required notions of limit superior and sublinearity for IVFs is provided. Definition 3.1. (Supremum and limit superior of an IVF). Let S be a nonempty subset of X and F : S → I(R) be an IVF. Then, the supremum of F over S is defined by The limit superior of F at a limit pointx in S is defined by Definition 3.2. (Infimum and limit inferior of an IVF). Let S be a nonempty subset of X and F : S → I(R) be an IVF. Then, the infimum of F is defined by The limit inferior of F at a limit pointx in S is defined by Lemma 3.1. Let S be a nonempty subset of X and F, G : S → I(R) be two IVFs. Then, at anȳ x ∈ S, the following properties are true: exists finitely, then the limit superior value is called upper gH-Clarke derivative of F atx in the direction h, and it is denoted by F C (x)(h). If this limit superior exists for all h ∈ X , then F is said to be upper gH-Clarke differentiable atx.
Definition 3.4. (Lower gH-Clarke derivative). Let F be an IVF on a nonempty subset S of X . Forx ∈ S and h ∈ X , if the limit inferior exists finitely, then the limit inferior value is called lower gH-Clarke derivative of F atx in the direction h. If this limit inferior exists for all h ∈ X , then F is said to be lower gH-Clarke differentiable atx.
If F has both upper and lower gH-Clark derivatives at x and they are equal, then F is called gH-Clark differentiable at x.

Remark 3.
Conventionally, for real valued-functions, the terminologies Clarke derivative [7,21] and upper Clarke derivative [9] are interchangeably used. In fact, the upper Clarke derivative is usually referred to as Clarke derivative. However, in order to avoid any confusion, we prefix upper and lower with the Clarke derivative corresponding to the values given by limit superior and limit inferior, respectively. In addition, throughout the article, we use the notation F C to refer the upper gH-Clarke derivative of an IVF F. Further, Then, from the inequalities (2) and (3), we get F C (x)(h) = |h| ⊙ C. Proof. See Appendix E.
Remark 5. Let S be a nonempty subset of X and the IVF F : S → I(R) be upper gH-Clarke diffrentiable atx ∈ S. Then, F may not be gH-directional differentiable atx ∈ S. For example, take X as the Euclidean space R, S = X and the IVF F : S → I(R), which is defined by Hence, F is upper gH-Clarke differentiable atx = 0. However, the limit does not exist atx = 0. Consequently, F is not gH-directional differentiable atx.
Remark 6. Let S be a nonempty subset of X and F : S → I(R) has gH-directional derivative at x ∈ S. Then, F is not necessarily upper gH-Clarke differentiable atx ∈ X . For instance, take X as the Euclidean space R 2 , S = {(x 1 , x 2 ) ∈ R 2 : x 2 ≥ 0, x 2 ≥ 0} and the IVF F : S → I(R), which is defined by Then, atx = (0, 0) and h = (h 1 , h 2 ) ∈ X such that for sufficiently small λ > 0 so thatx+λh ∈ S, we have Hence, F has a gH-directional derivative atx in every direction h ∈ X . Again, for x = (x 1 , x 2 ) ∈ S and h = (h 1 , h 2 ) ∈ X , we have This implies that F has no upper gH-Clarke derivative atx ∈ S.
The following theorem extends the well-known result from [21] for Lipschitz continuous functions to gH-Lipschitz continuous IVFs with the help of Lemma 3.2.
Theorem 3.1. Let S be a nonempty subset of X withx ∈ int(S) and F : S → I(R) be a gH-Lipschitz continuous IVF atx with a Lipschitz constant K ′ . Then, F is upper gH-Clarke differentiable atx and Proof. Since F is gH-Lipschitz continuous on S, for any h ∈ X , we get for λ > 0 that if x and λ are sufficiently close tox and 0, respectively. From inequality (4) we have Hence, the limit superior f C (x)(h) and f C (x)(h) exist atx (cf. p. 69 of [21]). By Lemma 3.2, the limit superior F C (x)(h) exists. Furthermore, by gH-Lipschitz continuity of F on S, we have the following for all h ∈ X : (4).
For convex and gH-Lipschitz continuous IVFs, upper gH-Clarke derivative and gH-directional derivative coincide as the next theorem states.
Theorem 3.2. Let X be convex, and the IVF F : X → I(R) be convex on X and gH-Lipschitz continuous at somex ∈ X . Then, the upper gH-Clarke derivative of F atx coincides with the gH-directional derivative of F atx in the direction h ∈ X .
Proof. Since F is a convex IVF on X , we get by Theorem 3.1 of [16] that the gH-directional derivative of F exists atx ∈ X in every direction h. Also, as F is gH-Lipschitz continuous atx, from Theorem 3.1, we get that the upper gH-Clarke derivative of F exists at anyx ∈ X in every direction h. Thus, by Definitions 2.4 and 3.3, we observe that For the proof of the reverse inequality, we write Since F is convex on X , Lemma 3.1 of [16] leads to the equality and for an arbitrary α > 0, Because of the gH-Lipschitz continuity of F atx, we have for sufficiently small ǫ > 0 and Then, by (ii) of Lemma 2.2, we have Since α > 0 is chosen arbitrarily, we obtain From (5) and (6), we get Example 3.2. Let X be the Euclidean space R and S = X . Then, the IVF F : S → I(R) that is defined by is sublinear on S. The reason is as follows.
For all x, y ∈ S and λ ≥ 0, we have Hence, F is a sublinear IVF on S. Example 3.3. Let Q be a real positive definite matrix of order n × n and S be a linear subspace of X . Consider the IVF F : S → I(R), which is defined by Then, F is a sublinear IVF on S. The reason is as follows.
The function F(x) can be written as g(x) ⊙ C, where g(x) = x T Qx. By Example 1.2.3 of [18], g satisfies the following conditions: (a) for λ ≥ 0 and x ∈ S, and (b) for all x, y ∈ S g(x + y) ≤ g(x) + g(y).
From (7), we have Since C ⊀ 0, from (8) and Lemma 2.1, we obtain Hence, F is a sublinear IVF on S.
Example 3.4. Let S be a linear subspace of X and F : S → I(R) be a convex IVF on S such that for all x ∈ S, F(αx) = α ⊙ F(x) for every α ≥ 0.
Then, F is a sublinear IVF on S. The reason is as follows. For x, y ∈ S and λ 1 , λ 2 > 0, we have Taking λ 1 = λ 2 = 1, we obtain Hence, F is a sublinear IVF on S.

Remark 7.
A sublinear IVF may not be convex. For instance, take X as the Euclidean space R, S = X and the IVF F : S → I(R) that is given by Clearly, by Example 3.2, F is a sublinear IVF on S. However, f (x) = −3|x| is not convex on S. Therefore, by Lemma 2.3, F is not a convex IVF on S. Theorem 3.3. Let S be a subset of X with nonempty interior, and let F : S → I(R) be an IVF that is upper gH-Clarke differentiable atx ∈ int(S). Then, the upper gH-Clarke derivative F C (x) of F is a sublinear IVF on S.
Proof. For an arbitrary h ∈ S and α ≥ 0, we have Next, for all h 1 , h 2 ∈ S, we get Hence, F C (x) is a sublinear IVF on S.

Conclusion and Future Directions
In this article, mainly three concepts on IVFs have been studied-limit superior of IVF (Definition 3.1), upper gH-Clarke derivative (Definition 3.3), and sublinear IVF (Definition 3.5). One can trivially notice that in the degenerate case, each of the Definitions 3.1, 3.3, and 3.5 reduces to the respective conventional definition for the real-valued functions. It has been observed that for a gH-Lipschitz continuous IVF, the upper gH-Clarke derivative always exists (Theorem 3.1). Also, for a gH-Lipschitz continuous IVF, it has been found that the gH-directional derivative of a convex IVF coincides with the upper gH-Clarke derivative (Theorem 3.2). It has been noticed that the upper gH-Clarke derivative at an interior point of the domain of an IVF is a sublinear IVF (Theorem 3.3).
In analogy to the current study, future research can be carried out for other generalized directional derivatives for IVFs, e.g., Dini, Hadamard, Michel-Penot, etc., and their relationships [9].
In parallel to the proposed analysis of IVFs, another promising direction of future research can be the analysis of the fuzzy-valued functions (FVFs) as the alpha-cuts of fuzzy numbers are compact intervals [15]. Hence, in future, one can attempt to extend the proposed idea of gH-Clarke derivative for fuzzy-valued functions.
The applications of the proposed upper gH-Clarke derivative in control systems and differential equations in a noisy or uncertain environment can be also dealt with in the future. A control system or a differential equation in noisy environment inevitably appears due to the incomplete information (e.g., demand for a product) or unpredictable changes (e.g., changes in the climate) in the system. The general control problem in a noisy or uncertain environment that we shall consider to study is the following: Here, H and F are upper gH-Clarke and gH-Fréchet differentiable IVFs with respect to the control variable u. In such a control problem, we shall show the usefulness of the proposed upper gH-Clarke derivative to find the optimal control of the system.
(i) We have the following four possible cases.
If possible, let which is an impossibility.
• Case 4. Let a − c > a − c and c − b < c − b. Proceeding as in Case 3 of (i) we can prove that (B.1) is not true. Hence, (ii) As B ⊖ gH A I(R) = max{|b − a|, |b − a|}, we break the proof in two cases. (iii) If possible, let there exist A, B, C and D in I(R) such that According to the definition of gH-difference of two intervals, Then, one of the following holds true: which is again impossible.
For this case, two subcases are similar to the Case 1 of (iii) will lead to impossibilities.
which is again contradictory to (B.9).
All the two subcases for this case are similar to Case 3 of (iii).
Hence, (B.6) is wrong, and thus the result follows.
Appendix C. Proof of Lemma 2.4 Proof. (i) Let F be gH-continuous atx ∈ S. Thus, for any d ∈ R n such thatx + d ∈ S, i.e., f and f are continuous atx ∈ S.
Conversely, let the functions f and f be continuous atx ∈ S. If possible, let F be not gHcontinuous atx. Then, as d → 0, (F(x + d) ⊖ gH F(x)) → 0. Therefore, as d → 0 at least one of the functions (f (x + d) − f (x)) and (f (x + d) − f (x)) does not tend to 0. So it is clear that at least one of the functions f and f is not continuous atx. This contradicts the assumption that the functions f and f both are continuous atx. Hence, F is gH-continuous atx.
(ii) Let F be gH-Lipschitz continuous on S. Thus, there exists K > 0 such that for any x, y ∈ X we have Hence, f and f are Lipschitz continuous on S.
Conversely, let the functions f and f be Lipschitz continuous on S. Thus, there exist K 1 , K 2 > 0 such that for all x, y ∈ S, Hence, F is gH-Lipschitz continuous IVF on S.
(iii) Let F be gH-Lipschitz continuous on S. Then, there exists an K > 0 such that for all x, y ∈ S, we have F(y) ⊖ gH F(x) I(R) ≤ K y − x .
Hence, F is upper gH-Clarke differentiable IVF atx ∈ S.